where \({\mathbf {\theta }}\in \mathbb {S}^{d-1}\)  is the mean direction of the common class distribution.
Note that the last equation integrates over all possible values of \(\mathbf {z}_i\) .
Assuming that proposals are i.i.d samples from their
corresponding distributions given their labels \(\mathbf {z}_i\)i.i.d assumption is a standard approach in MIL and works well in practice. See [1]} for more details., we can write
\(p(F_i|\mathbf {z}_i=\mathbf {e}_j, {\mathbf {\theta }}) = p_{{\mathbf {\theta }}}^+(F_{ij})\prod _{\begin{array}{c}k=1\\k\ne j\end{array}}^P p_{{\mathbf {\omega }}}^-(F_{ik})~,\) 
